Section 4.2 Adding, Subtracting and Multiplying Polynomials Honors Algebra 2 Section 4.2 Adding, Subtracting and Multiplying Polynomials

Warm Up Do the problems in Exploration 1 on page 165

Adding and Subtracting Polynomials Your book has two methods. Skip the vertical method! If you know how to do distributive property and combine like terms, you can add and subtract polynomials!! Always write final answers in standard form.

#1 Add 2 𝑥 3 + 𝑥 2 −3𝑥+4 and 5 𝑥 3 −7 𝑥 2 −8 #2 Find the sum #1 Add 2 𝑥 3 + 𝑥 2 −3𝑥+4 and 5 𝑥 3 −7 𝑥 2 −8 #2 Find the sum. 2𝑥 4 −6 𝑥 3 +9 𝑥 2 −𝑥+1 +( 𝑥 5 + 5𝑥 3 −6𝑥+2) #3 Subtract 5𝑥 3 − 2𝑥 2 +𝑥−7 from 4 𝑥 3 +2 𝑥 2 +10 #4 Find the difference. 2𝑥 3 − 8𝑥 2 +4𝑥−11 −( 𝑥 4 + 4𝑥 3 −6𝑥+2)

Multiplying Polynomials Case#1 Monomial times Polynomial Use distributive property! 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 3 𝑥 2 (4 𝑥 4 −3 𝑥 2 +5)

Multiplying Polynomials Case #2 Polynomial times Polynomial For binomials, use FOIL. #1 3𝑥+5 2𝑥−7 #2 ( 𝑥 2 −8)( 𝑥 2 −2) #3 𝑦+6 2𝑦+9 #4 (𝑚−1)(𝑚+1)

Multiplying Polynomials Any polynomial × Any polynomial Count the number of terms in the first factor Count the number of terms in the second factor The product of the counted terms is the number of multiplications to be done.

𝑥−3 𝑥 2 +3𝑥−7 Six multiplications 𝑥( 𝑥 2 )+𝑥 3𝑥 −𝑥 7 −3( 𝑥 2 )−3 3𝑥 −3(−7) 𝑥 3 +3 𝑥 2 −7𝑥−3 𝑥 2 −9𝑥+21 Now combine like terms! 𝑥 3 −16𝑥+21 You can also use a multiplication table and add like boxes.

Using a table to multiply 𝑥 3 +6 𝑥 2 +𝑥−14

Multiply #1 ( 𝑥 2 −5𝑥+2)(𝑥−6) #2 ( 𝑥 2 +3𝑥−10)(2 𝑥 2 −4𝑥+1) You can use a table or just multiply terms! Make sure you do all the required multiplications! TRY THESE!

How can you multiply more than two polynomials? What do you do to multiply several numbers? Find the product: (2𝑥−1)(𝑥+2)(3𝑥−4)

Special Cases Sum and Difference 𝑎+𝑏 𝑎−𝑏 = 𝑎 2 − 𝑏 2 𝑥−4 𝑥+4 = 𝑥−4 𝑥+4 = Square of a binomial (𝑎+𝑏) 2 = 𝑎 2 +2𝑎𝑏+ 𝑏 2 (𝑥+3) 2 = (𝑎−𝑏) 2 = 𝑎 2 −2𝑎𝑏+ 𝑏 2 (𝑚−5) 2 =

Cube of a binomial (𝑎+𝑏) 3 = 𝑎 3 + 3𝑎 2 𝑏+ 3𝑎𝑏 2 + 𝑏 3 (𝑝+3) 3 = (𝑎−𝑏) 3 = 𝑎 3 − 3𝑎 2 𝑏+ 3𝑎𝑏 2 − 𝑏 3 (𝑧−6) 3 =

Try these! Find the products! #1 4𝑐+5 4𝑐−5 #2 (3𝑚−1) 2 #3 (𝑟𝑠+2) 3

What if you had to find the following (𝑥+2) 6 Any ideas?

Pascal’s Triangle can help to do a binomial raised to a power

Think about the binomials we raised and look for a relationship with Pascal’s Triangle

Use Pascal’s Triangle to expand the following. (2𝑟−1) 5

Assignment #16 Pg. 170 #3,7,9,13,17-21 odd,25,29,34,35-41 odd, 45,47,51,56